Diagonalizable matrix and Zariski topology

Question

Identify the set of square matrices of order two defined over $ \mathbb{C}$ with the affine space $A^4$. What can you tell about the set of diagonalisable matrices? I noticed that it corresponds to a union of an open Zariski set and a closed Zariski set. It is this set open or closed?

Answer 1

It is neither.

• Since it contains a non-empty Zariski open set (for instance, the set of matrices with $n$ dinstinct eigenvalues), it is dense in $\Bbb A^4$, but not all matrices are diagonalisable.

• The set of non-diagonalisable matrices is not Zariski closed, because it isn't euclidean-closed. For instance $$\begin{pmatrix}1&\varepsilon&0&0\\ 0&1&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}\stackrel{\varepsilon\to0}\longrightarrow \begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}$$